If alpha,beta and gamma rays carry same momentum, which has the shortest wavelength ?

Options

A. alpha rays
B. beta rays
C. gamma rays
D. none all have same wavelength.

Explanation

To solve the problem of determining which of the alpha, beta, and gamma rays has the shortest wavelength when they all carry the same momentum, we can use the de Broglie wavelength formula.

### Step-by-Step Solution:

1. **Understand the de Broglie Wavelength Formula**:
The de Broglie wavelength (λ) of a particle is given by the formula:
\[ \lambda = \frac{h}{p} \]
where \( h \) is Planck's constant and \( p \) is the momentum of the particle.

2. **Identify the Given Information**:
We are given that alpha (α), beta (β), and gamma (γ) rays all have the same momentum. This means:
\[ p_{\alpha} = p_{\beta} = p_{\gamma} \]

3. **Apply the de Broglie Wavelength Formula for Each Ray**:
Using the de Broglie formula, we can express the wavelengths for each type of ray:
- For alpha rays:
\[ \lambda_{\alpha} = \frac{h}{p_{\alpha}} \]
- For beta rays:
\[ \lambda_{\beta} = \frac{h}{p_{\beta}} \]
- For gamma rays:
\[ \lambda_{\gamma} = \frac{h}{p_{\gamma}} \]

4. **Substitute the Equal Momentum**:
Since \( p_{\alpha} = p_{\beta} = p_{\gamma} \), we can substitute this into the equations:
\[ \lambda_{\alpha} = \frac{h}{p}, \quad \lambda_{\beta} = \frac{h}{p}, \quad \lambda_{\gamma} = \frac{h}{p} \]
where \( p \) is the common momentum.

5. **Conclusion**:
Since all three wavelengths are equal:
\[ \lambda_{\alpha} = \lambda_{\beta} = \lambda_{\gamma} \]
Therefore, none of the rays has a shorter wavelength than the others; they all have the same wavelength.

### Final Answer:
All three rays (alpha, beta, and gamma) have the same wavelength.